|
%I #5 Dec 05 2018 07:58:41
%S 1,4,6,8,10,12,14,15,16,18,20,22,24,26,28,30,32,33,34,35,36,38,40,42,
%T 44,45,46,48,50,51,52,54,55,56,58,60,62,64,66,68,69,70,72,74,75,76,77,
%U 78,80,82,84,85,86,88,90,92,93,94,95,96,98,99,100,102,104
%N Heinz numbers of disconnected integer partitions.
%C Differs from A289509 in having 1 and lacking 2, 195, 455, 555, 585...
%C Also positions of entries > 1 in A305079.
%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C An integer partition is connected if the prime factorizations of its parts form a connected hypergraph. It is disconnected if it can be separated into two or more integer partitions with relatively prime products. For example, the integer partition (654321) has three connected components: (6432)(5)(1).
%e The sequence of all disconnected integer partitions begins: (11), (21), (111), (31), (211), (41), (32), (1111), (221), (311), (51), (2111), (61), (411), (321), (11111), (52), (71), (43), (2211), (81), (3111), (421), (511), (322), (91), (21111), (331), (72), (611), (2221), (53), (4111).
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t Select[Range[200],Length[csm[primeMS/@primeMS[#]]]>1&]
%Y Cf. A054921, A218970, A286518, A290103, A304714, A304716, A305078, A305079, A322306, A322307, A322338, A322367, A322369.
%K nonn
%O 1,2
%A _Gus Wiseman_, Dec 04 2018
|
